A hybrid framework for compartmental models enabling simulation-based inference

Multi-scale systems often exhibit stochastic and deterministic dynamics. Capturing these aspects in a compartmental model is challenging. Notably, low occupancy compartments exhibit stochastic dynamics and high occupancy compartments exhibit deterministic dynamics. Failing to account for stochasticity in small populations can produce 'atto-foxes', e.g. in the Lotka-Volterra ordinary differential equation (ODE) model. This limitation becomes problematic when studying extinction of species or the clearance of infection, but it can be resolved by using discrete stochastic models e.g. continuous time Markov chains (CTMCs). Unfortunately, simulating CTMCs is infeasible for most realistic populations. We develop a novel mathematical framework, to couple continuous ODEs and discrete CTMCs: 'Jump-Switch-Flow' (JSF). In this framework populations can reach extinct states ("absorbing states"), thereby resolving atto-fox-type problems. JSF has the desired behaviours of exact CTMC simulation, but is substantially faster than existing alternatives. JSF's utility for simulation-based inference, particularly multi-scale problems, is demonstrated by several case-studies. In a simulation study, we demonstrate how JSF can enable a more nuanced analysis of the efficacy of public health interventions. We also carry out a novel analysis of longitudinal within-host data from SARS-CoV-2 infections to quantify the timing of viral clearance. JSF offers a novel approach to compartmental model development and simulation.